How could I animate a plane into a pipe and then a pipe into a torus?
I would probably do this with 2 'Simple Deform' modifiers set to 'Bend' and animate the 'Angle' value for both:



I start off with a subdivided plane (with it's scale and rotation applied: Ctrl+A):



I add a 'Simple Deform' modifier, set it to 'Bend' and set the 'Angle' to '360'. Because the bend isn't going in the right direction I use an empty (Shift+A> Empty), placed at the object's origin, and give it a rotation of 90 degrees on it's 'x' axis. Setting this empty as the 'Simple Deform' 'Origin' corrects the bend orientation:



I then repeat the previous step so the plane now has 2 'Simple Deform' modifiers, and again, add an empty. This time I rotate the empty '90' degrees on it's 'x' axis and '90' on it's 'z' axis.



You can then animate the 'Angle' properties of both 'Simple Deform' modifiers to create the animation by pressing I over the 'Angle' properties at values between '0' and 360', animating the first modifier and then the second. 

I also checked 'double sided' on the object data properties for the plane, otherwise the shading will look odd as essentially the mesh is bending into an 'inside-out' position:



If you want to change where the 'seam' of the roll up is you can edit the position of the plane in edit mode relative to the object origin. 

Here is the blend file: